You're running late for your flight at the airport and have to get to the gate before it closes. Your journey is partly just walking and partly walking on a travellator. You notice your shoelace has come undone. Is it better to stop to tie your shoelace while you're on the travellator, or off it, or doesn't it make any difference?

Auxiliary similar question, similar scenario: you have the energy to put on a brief burst of speed but you're not fit enough to maintain it for long. Is it better to speed up your walking on the travellator, or off it, or doesn't it make any difference?

I heard this one on the BBC "Today" morning news program. I figured out the answer and when googling to check I'd got it right (BBC wait till the next day before giving the solution) I found the auxiliary question and that it was originally devised by Terence Tao, famous mathematician and Fields Medal winner.

I got kind of caught up in side details. One, it was obvious in context what it is, but I had never heard the word travelator before just now, and is this British, or just a term I somehow have missed all my life? What did I call those things before just now? I forgot. Walkways or something?

I did get the right answer, but only after I assumed that it was a pretty straight question, but my first thought was:

Do you go more slowly with an untied shoe, or risk falling down and getting hopelessly behind? Then you should tie your shoe right away. If neither of those is an issue, then why stop to tie your shoe at all? Just do it later after you're safely at the gate.

I guess 'travellator' (apparently can be spelt with either one or two l's) is a British term for a moving walkway. Obviously derived from 'escalator' which I think IS used in the USA as well as Britain?

I accept lisa's quibble, but it is only a puzzle If your shoelace gets tangled in the end of the travellator where it goes back underground it might rip your shoe off or injure your foot which would really slow you down! Or feel free to invent your own reason for briefly stopping walking. There is an auxiliary similar question under the first spoiler tag.

I figured out it was a straight math question and not a lateral thinking question or a more detailed solution because they didn't provide enough detail for it to be either of those.

But this is a longstanding beef I have with that sort of thing, where puzzles add extraneous details for realism or something, and I have to stop to consider why all that stuff is even there. I also get really suspicious when people ask me really easy questions, because I always think it's probably a trick so I waffle and have to think really hard, and it makes me look like the stupidest possible guy.

In the US we call them people movers. Since they are the only thing that moves people...

It seems like you'd want to tie your shoe on the travellator. But taking the obvious answer always makes me think I am missing something. Though it ignores the fact that other people on the travellator might possibly trip over you or kick you.

The other people on the travellator are only going at walking speed relative to you if you stop - so they don't have much of an excuse to trip over you or kick you other than the travellator probably being narrower than the normal airport corridors.

What about the best place to briefly speed up? Apparently, when asked to make a quick guess at the answer to that one, more people choose each of the two wrong answers than choose the right one.

IF we presume that the length of the travellator is such that you will finish tying your shoe before you reach the end of the travellator.

If you stop off the travellator, you add the shoe-tying time on, you don't spend any less time walking or riding the travellator.

If you stop on the travellator, you will spend the same time walking, and you will spend some time going slower on the travellator, but you won't waste the whole time not moving at all.

The important factor for the second question is that the travellator is a fixed length not a fixed time. If instead the travellator were the entire walkway, but it only had two minutes of power and the question was whether to sprint while the power was on, or after the power ran out, I believe it wouldn't make any difference (for a sufficiently long travellator).

But since it's a fixed length, you sprinting on the travellator means that you also spend less time on the travellator, so you get less time getting the machine-aided speed boost.

The answers are more obvious if you increase the ratios in the speeds and such to an unrealistic level. Suppose the travellator goes at 60 mph and travels for a mile. You also will walk a half mile off the travellator, taking 10 minutes (so 3 mph). It takes you 1 minute to tie your shoe. So you can spend the entire time on the travellator tying your shoe, and take a total of 11 minutes. Or you can spend 10 minutes walking and 1 minute tying your shoe (uh-oh, you've already equaled the time from the other way) and THEN get on the travellator and go 63 mph and take slightly less than a minute... for a total of 11 min ~57 sec.

If you walk the whole way, you spend 10 min 57 sec then.

Taking a burst of speed, you could run 9 mph for 30 seconds... Well, you could do that off of the travellator, and save 1 min, for 9 min 57 sec. But you already know that you only spend 1 minute on the travellator if you don't move at all! And by walking on it, from the previous example, you'll spend ~57 seconds since you'd be going 63 mph. So you know that you can't possibly save more than 1 minute by running on the travellator when walking takes you less than a minute. In fact, you'll spend ~54 seconds on the travellator instead of 57 sec, for a total of 10 min 54 sec, saving you 3 seconds vs. not sprinting at all and just walking the whole time.

I wrote all of this before googling it, JUST SO YOU KNOW. Except this last part, where I talk about googling it. Everything before this paragraph.